200
4 Numerical Methods and Simulation for Pebble Flows
Fig. 4.23 The multi-fractal analysis for all recirculation flow rates. The scaling exponent K(q) is
according to Eqs. (11–12), for q = 1 to 6. a F1. b F2. c F3. d F4. e F5. fF6. g The plot of K(q)
versus q for all recirculation flow rates
Table 4.9 The fitting parameters for Fig. 4.23
Equation
y = a + bx
q
q = 1
q = 2
q = 3
q = 4
q = 5
q = 6
F1
a
−4.410
−2.080
1.039
4.330
7.688
11.081
b
0
−0.646
−1.3567 −2.086
−2.824
−3.567
R-square 1
0.98272
0.9844
0.9852
0.98567
0.98605
F2
a
−0.175
2.75991
6.48512
10.66579 15.14776 19.82144
b
0
−0.34372 −0.76368 −1.23191 −1.73418 −2.25792
R-square 1
0.96932
0.97564
0.98107
0.98471
0.98699
F3
a
0.00571
1.61808
4.08906
7.09075 10.4303
13.98851
b
0
−0.23075 −0.57569 −0.98806 −1.4416 −1.92026
R-square 1
0.98032
0.97895
0.97894
0.9793
0.97946
F4
a
−0.00183 1.08322
2.8431
5.03979
7.52371 10.20009
b
0
−0.16389 −0.42407 −0.74396 −1.1011 −1.4815
R-square 1
0.97686
0.9788
0.97982
0.9805
0.98104
F5
a
0.0019
0.62819
1.63687
2.9213
4.41825
6.08187
b
0
−0.09838 −0.25338 −0.44772 −0.67159 −0.91792
R-square 1
0.92868
0.9377
0.94298
0.94631
0.94856
F6
a
0.0511
0.64759
1.57726
2.7492
4.112
5.62947
b
0
−0.08643 −0.22238 −0.39316 −0.59085 −0.80997
R-square 1
0.92018
0.93129
0.93777
0.94171
0.94417